The Saudi government in recent days has released at least two dozen high-profile suspects held in a wide-ranging crackdown on corruption, a sign that those accused of illegally amassing wealth are increasingly agreeing to settle as authorities push to expedite the investigation process.

Among those released in the past week is Ibrahim al-Assaf, a former finance minister and board member of the state oil giant Saudi Aramco, who was accused of embezzlement related to the expansion of Mecca’s Grand Mosque and taking advantage...